Greene Ramos

08/26/2024 · High School

A decision maker's utility function is given by \( u(w)=-e^{-5 w} \). The decision maker has two random economic prospects (gains) available. The outcome of the first, denoted by \( X \), has a normal distribution with mean 5 and variance 2 . Henceforth, a statement about a normal distribution with mean \( \mu \) and variance \( \sigma^{2} \) will be ab- breviated as \( N\left(\mu, \sigma^{2}\right) \). The second prospect, denoted by \( Y \), is distributed as \( N(6,2.5) \). Which prospect will be preferred? a. Assume \( u^{\prime \prime}(w)<0, \mathrm{E}[X]=\mu \), and \( \mathrm{E}[u(X)] \) exist; prove that \( \mathrm{E}[u(X)] \leq u(\mu) \). [Hint: Express \( u(w) \) as a series around the point \( w=\mu \) and terminate the expansion with an error term involving the second derivative. Note that Jen- sen's inequalities do not require that \( u^{\prime}(w)>0 \).] b. If \( u^{\prime \prime}(w)>0 \), prove that \( \mathrm{E}[u(X)] \geq u(\mu) \). c. Discuss Jensen's inequalities for the special case \( u(w)=w^{2} \). What is \( \mathrm{E}[u(X)]-u(\mathrm{E}[X]) \) ?